Aubry Duality and a Thouless formula forquasi-periodic Schrödinger difference

equations

Àlex Haro1 & Joaquim Puig2

Grup de Sistemes Dinàmics UB-UPC(1) Departament de Matemàtica Aplicada i Anàlisi (UB).

(2) Departament de Matemàtica Aplicada I (UPC).

New Perspectives in Discrete Dynamical SystemsTossa de Mar, October 4th 2014

Quasi-periodic Schrödinger operatorsIn many physically relevant situations (eg. quasi-crystals,Quantum Hall Effect, linearization around q-p orbits in DS) it isnecessary to consider a Schrödinger operator withquasi-periodic potential:(

Hb,ω,φx)

n = xn+1 + xn−1 + bwnxn.

where wn = (W (ωn + φ))n∈Z is a quasi-periodic sequence withI W : T = R/Z→ R a real analytic funcion, called the

potential.I ω ∈ R a irrational frequency,I φ ∈ T a phase.I b is a coupling parameter.

and any of these operators satisfiesI It is bounded and self-adjoint on l2(Z).I The spectrum is independent of φ since ω is irrational.

The dynamical connection: the eigenvalue equationA link between spectral theory and dynamical systems for thisequation is to consider the corresponding eigenvalue equation:

xn+1 + xn−1 + bW (nω + φ)) xn = axn, n ∈ Z

where a is the energy. These are discrete difference equationsand quasi-periodic versions of the classical Hill’s equation

x ′′(t) + (a + bq(t)) x(t) = 0, q(t) = q(t + T ).

A “naive” discrete analog is the Harper equation

xn+1 + xn−1 + b cos 2π (nω + φ) xn = axn, n ∈ Z,

the eigenvalue equation of the Almost Mathieu operator(Hb,ω,φx

)n = xn+1 + xn−1 + b cos 2π (nω + φ) xn.

A dynamical perspectiveThese ev equations can be viewed as linear skew-products:(

xn+1xn

)︸ ︷︷ ︸

vn+1

=

(a− bW (θn) −1

1 0

)︸ ︷︷ ︸

Aa,b(θn)

(xn

xn−1

)︸ ︷︷ ︸

vn

,

θn+1 = θn + ω (mod 1),

and the solution is given by a cocycle on SL(2,R)× T

M(N)a,b,ω(θ0) =

Ma,b (θN−1) . . .Ma,b(θ0) N > 0,I N = 0,M−1

a,b (θN) . . .M−1a,b(θ−1) N < 0.

(Aa,b,2πω

)n(I, θ0) =

(M(N)

a,b,ω(θ0), θ0 + nω)

(Aa,b, ω) : SL(2,R)× T −→ SL(2,R)× T(X , θ) 7→

(Aa,b(θ)X , θ + ω

).

Another perspective (not new, though. . . )

Writing yn = xn−1/xn, a family of Harper-like maps on R× Tyn+1 =

1a− bW (θn)− yn︸ ︷︷ ︸

fa,b(yn,θn)

,

θn+1 = θn + ω (mod 1),

I y ∈ R = [−∞,+∞] and θ ∈ T = R/Z.I b (coupling), a (energy) and ω (irrational frequency) are

parameters. Harper happens when W = cos.A Harper map is a skew-product map on R× T

Fa,b,ω(yn, θn) = (fa,b(yn, θn), θn + ω),

F (N)a,b,ω(y0, θ0) = (f (N)

a,b (y0, θ0), θ0 + Nω), N ∈ Z.

. . . which is an qpf circle mapTo get rid of the point at∞, take polar coordinates ϕ = arctan yso that y ∈ P ' [−π/2, π/2] and the resulting equations are

ϕn+1 = arctan(

1a− b cos (2πθn)− tanϕn

)︸ ︷︷ ︸

fa,b(ϕn,θn)

,

θn+1 = θn + ω (mod 1).

Since P× T ' S1, it is a quasi-periodically forced circle map.Take home message of the talk

Quasi-periodic Schrödinger operators (and their differenceequations, skew-producs and maps)

I Display interesting and nontrivial phenomena: coexistenceof different spectral types, nonuniform hyperbolicity orSNAs . . .

I Their study requires the combination of different areas.

An excursion in Etymology

SNAStrange Nonchaotic Attractor

An excursion in Etymology

ANCEAtractor No Caòtic Estrany

An excursion in Etymology

ANCEAtractor No Caótico Extraño

An excursion in Etymology

DANCE

The DANCE Network: History

≈2000 Ll. Alsedà and A. Delshams became interested in SNA2001 XT2001: Application for a Catalan network "Dinàmica

discreta en dimensió baixa i atractors estranys" (DiscreteDynamics in low dimension and strange attractors), formedby 5 nodes (UAB, UB, UGR, UOV and UPC), unsuccessful.

2002 BFM2001: Spanish network DANCE (Dinámica no linealen dimensión baja y atractores extraños), formed by 7nodes (UAB, UB, UGR, UOV, UPC, US and UVA),successful!!!

2003 Ddays 2003 in Salou: Total discussion about SNA2003 BFM2002: Spanish network DANCE (Dinámica, Atractores

y Nolinealidad: Caos y Estabilidad), formed by 10 nodes(UAB, UB, UGR, UIB, UM, UOV, UPC, US, UV and UVA)

2004 RTNS 2004 in Palma de Mallorca

The DANCE network today

Today The DANCE network (http://www.dance-net.org/) has 21nodes, more than 200 researchers

I Ddays 2003, 2004, 2006, 2008, 2010, 2012, 2014I RTNS 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011,

2012, 2013, 2014I There have been several other coordinators, but Lluís

Alsedà is currently again one of the two coordinatorsI The SNA topic has been studied by several people of the

group: Lluís Alsedà, Sara Costa, Jordi-Lluís Figueras, ÀlexHaro, Àngel Jorba, Carmen Núñez, Rafael Obaya,Joaquim Puig, Pau Rabassa, Joan Carles Tatjer, . . . .

DANCE rules

I Try new dances.I Do not dance alone.

Lyapunov exponent. Almost Mathieu & beyond(Upper) Lyapunov exponent of xn+1 + xn−1 + bW (θn)xn = axn

γH(a,b) = limN→∞

1N + 1

∫T

log∥∥Aa,b(2πNω + θ) · · ·Aa,b(θ)

∥∥dθ

well-studied behaviour for the AMO W (θ) = cos 2πθ. In thespectrum it equals max

(0, log |b|2

). For b = 1,2,4:

0

2

4

6

8

0 2 4 6 8

Towards an explanation for the AMO: the IDS

κL,b,ω,φ(a) =1L

#

eigenvalues ≤ a of Hb,ω,φ|1,...,L

for fixed with a, b and φ and some boundary conditions. Then

limL→∞

κL,b,ω,φ(a) = kb,ω(a),

is the integrated density of states (IDS), exists and satisfiesI it is independent of φ.I it is continuous and not decreasing function of a (b fixed).I a ∈ σ(b, ω), spectrum of Hb,ω,φ ⇔ the IDS increases at a.

One can recover the Lyapunov exponent from the IDS throughthe Thouless Formula, which holds for more general potentials:

γH(a,b) =

∫R

log |a− a′|dκb,ω(a′) for a ∈ C and b ∈ R.

Why is the Almost Mathieu, W = b cos, so special? I

The basic reason is invariance through Aubry DualityI Assume that a is a point eigenvalue of an AMO Hb,φ whose

e.f decays exponentially in |n| (homoclinic at zero).I This means that there is an exponentially decaying (ψn)n,

solution of the eigenvalue equation

ψn+1 + ψn−1 + b cos 2π(ωn + φ)ψn = aψn, n ∈ Z,

I Think of (ψn)n as the Fourier coefficients of an analyticfuncion on T and consider the quasi-periodic Bloch wave

xn = einφψ(ωn), ψ(θ) =∑k∈Z

ψkeikθ.

Why is the Almost Mathieu, W = b cos, so special? III This sequence (xn)n∈Z satisfies the difference equation

b2

(xn+1 + xn−1) + 2 cos(ωn)xn = axn,

an e.v. equation for the Almost Mathieu with parameters

β =4b, α =

2ab

I Aubry duality is this mechanism and applies to manysituations. For example, the IDS is invariant through duality

κ(a,b) = κ

(2ab,

4b

).

I Aubry duality can be made more precise underDiophantine conditions for ω and φ.

Duality of the Lyapunov exponent for the AMO

Using Thouless Formula for the Almost Mathieu

γH(a,b) =

∫R

log |a− a′|dκb,ω(a′)

and the duality of the IDS, a change of variables shows that

γH(a,b) = log|b|2

+ γH(

2ab,

4b

)Since βH is zero in the spectrum for |b| ≤ 2, (nontrivial butnatural, proved by Bourgain & Jitomirskaya 2002), in particular

γH(a,b) = max(

0, log|b|2

), a ∈ σ(b, ω).

Extension to more general potentials I

I The previous trick works only for the Almost Mathieu.I However, we can still use it for more general trigonometric

polynomials.I For example, consider a Schrödinger operator with a

potential with two harmonics:

(hx)n = xn+1 + xn−1 + 2β (cos (2πθn) + cos (4πθn))︸ ︷︷ ︸W (θn)

xn,

θn = θ0 + ωn, n ∈ Z,

with coupling β 6= 0.I W : T→ R could be any trigonometric polynomial.I We can compute numerically the Lyapunov exponents.

Extension to more general potentials II

3 2 1 0 1 2 30.2

0.0

0.2

0.4

0.6

0.8

1.0

Figure : Upper Lyapunov exponent for β = 0.25 and ω =√

5−12 .

Extension to more general potentials III

3 2 1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

Figure : Upper Lyapunov exponent for β = 0.5 and ω =√

5−12 .

Extension to more general potentials IV

3 2 1 0 1 2 30.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure : Upper Lyapunov exponent for β = 0.75 and ω =√

5−12 .

Extension to more general potentials V

3 2 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure : Upper Lyapunov exponent for β = 1 and ω =√

5−12 .

More numerical explorations

Questions

I For small values of β, the Lyapunov exponent is zero in thespectrum (well-understood by Eliasson’s reducibilitytheory).

I For large values of β, the Lyapunov exponent is alwayspositive ("easy" using Herman’s trick).

I For intermediate values, coexistence (cf. Avila ICM2014).I When the Lyapunov exponent is positive, it is not constant,

but rather on a smooth curve (related to the stratificatedregularity of the Lyapunov exponents (Avila 2013)).

I Is it possible to understand this through Aubry Duality?I Can this explanation be purely dynamical?

An Aubry Duality approach I

I The Aubry dual of the operator with two harmonics

(hx)n = xn+1 + xn−1 + 2β (cos (2πθn) + cos (4πθn))︸ ︷︷ ︸W (θn)

xn,

is a difference operator of order 4 (twice the deg. of W )

(h′x)n = β (xn+2 + xn+1 + xn−1 + xn−2)+2 cos(2πθn)xn, n ∈ Z.

I For any α, its eigenvalue equation h′x = αx , defines askew-product on R4,

xn+2xn+1xn

xn−1

=

−1 α

β −2β cos(2πθn) −1 −1

1 0 0 00 1 0 00 0 1 0

xn+1xn

xn−1xn−2

,

θn+1 = θn + ω.

An Aubry Duality approach II

I Unlike the Almost Mathieu, these linear skew-products arenot in SL(2,R) but they preserve an adapted complexsymplectic structure (dependent on W .)

I In particular, for any α ∈ C and β 6= 0, it has 2 Lyapunovexponents which are non-negative, γ1(α, β) and γ2(α, β).

I Which is the relationship between the Lyapunov exponentof the original Schrödinger operator h at α and these twoLyapunov exponents of the dual?

I We can show a similar relation than for the AMO:

γh(α, β) = γh′1 (α, β) + γh′

2 (α, β) + log |β|︸ ︷︷ ︸normalized Lyapunov semi-trace

, (1)

Numerical examples I

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W=2βcos(2π ·) +2βcos(4π ·), V=2cos(2π ·), β=0.25

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Numerical example

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W=2βcos(2π ·) +2βcos(4π ·), V=2cos(2π ·), β=0.5

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Numerical example

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W=2βcos(2π ·) +2βcos(4π ·), V=2cos(2π ·), β=0.75

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Numerical example

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W=2βcos(2π ·) +2βcos(4π ·), V=2cos(2π ·), β=1.0

3 2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Quasi-periodic long-range operatorsA quasi-periodic long-range operator of finite range d and(irrational) frequency ω ∈ R is a bounded self-adjoint operatorh = hV ,W ,θ0 acting on x = (xn)n ∈ `2(Z,C) by

(hx)n =d∑

k=−d

Vkxn+k + W (θn)xn, n ∈ Z,

whereI V : T→ R is a real trigonometric function with average 0,

the symbol, with Fourier representation

V (θ) =d∑

k=−d

Vke2πikθ;

I W : T→ R is a real analytic funcion, the potential;I θ0 ∈ T is a phase, and θn = θ0 + nω for n ∈ Z.

Quasi-periodic long-range linear skew-productsThe eigenvalue equation of h for α ∈ C,

d∑k=−d

Vkxn+k + W (θn)xn = αxn, n ∈ Z,

is equivalent to

xn+d...

xn+2xn+1

xn

.

.

.xn−d+2xn−d+1

︸ ︷︷ ︸

un+1

=1

Vd

−Vd−1 . . . −V1 α−W (θn) −V−1 . . . −V−d+1 −V−dVd

. . .Vd

VdVd

. . .Vd

︸ ︷︷ ︸

Ahα(θn)

xn+d−1...

xn+1xn

xn−1...

xn−d+1xn−d

︸ ︷︷ ︸

un

θn+1 = θn + ω,

defining a long-range linear skew-product(Ah

α, τ) : C2d × T→ C2d × T.

Symplectic properties of long-range cocyclesAn adapted complex symplectic structure

PropositionFor α ∈ R, the long-range skew-product (Ah

α, τ) is complexsymplectic with respect to the complex symplectic structure

Ω =

(0 −C∗

C 0

),

where

C =

Vd · · · V1

0. . .

...0 0 Vd

.

That is:1(Ah

α(θ))∗ Ω Ahα(θ) = Ω.

Compare with [Johnson 87].

Thouless formula for long-range operatorsGeneralization of previous objects

The integrated densitity of states of the operator h is anon-decreasing function κh : R→ [0,1] defined as the limit

κh(a) = limN→∞

12N + 1

#

eigenvalues ≤ a of h[−N,N],

where h[−N,N] is the restriction of h to the interval [−N,N] withzero boundary conditions.

The normalized Lyapunov semi-trace or entropy of h at α, γh(α)is

γh(α) = γh1 (α) + . . .+ γh

d (α) + log(Vd ),

where γh1 (α) ≥ . . . ≥ γh

d (α) ≥ 0 are the d non-negativeLyapunov exponents of (Ah

α, τ).

Thouless formula for long-range operatorsThe main result

Theorem (Thouless formula)The following integral formula holds:∫

Rlog |α− a|dκh(a) = γh(α),

where κh is the IDS of the long-range operator h, and γh(α) isthe normalized Lyapunov semi-trace.